Reverse mathematics, Young diagrams, and the ascending chain condition

Abstract

Let $S$ be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of $S$ over a field $K$ of characteristic $0$. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over RCA${}_0$ (or even over RCA${}_0^{\ast }$) to the statement that ${\omega }^{\omega }$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

arXiv comment14 pages